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# Fisika Matematika II (1 - 2) kalkulus-variasi

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### Fisika Matematika II (1 - 2) kalkulus-variasi

1. 1. Pertemuan 1 - 2KALKULUS VARIASI<br />Dr. I Made Astra, M.Si<br />JurusanFisika<br />FakultasMatematikadanIlmuPengetahuanAlam<br />1<br />
2. 2. Outline<br />Kalkulusvariasi : <br />persamaan Euler<br />persoalanbrachistochrome<br />persamaan Lagrange<br />persoalanisoperimetrik<br />07/01/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />2<br />
3. 3. Metode Euler<br />Menghitungpersamaandifferensialmelaluitaksiranlangsungdari slope xidiberiturunanpertama. <br />07/01/2011<br />3<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
4. 4. Metode Euler (Ex.)<br /><ul><li>Selesaikanpersamaandifferensial</li></ul>padainterval x = 0 s/d x = 1, h = ¼. Padasaat x = 0, nilai y = 1. Hitungkesalahansebenarnya!<br />07/01/2011<br />4<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
5. 5. Metode Euler (Ex.)<br />Untuk x = 0  y = 1<br />Untuk x = 0,25 <br />yi+1 = yi + f(xi, yi).h<br /> = 1 + f(0,1).0,25<br /> =<br /> = 1<br />Untuk x = 0,5 <br />yi+1 = yi + f(xi, yi).h<br /> = 1 + f(0,25;1).0,25<br /> = <br /> = 1,0625<br />07/01/2011<br />5<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
6. 6. Metode Euler (Ex.)<br />Untuk x = 0,75 <br />yi+1 = yi + f(xi, yi).h<br /> = 1,0625 + f(0,5;1,0625).0,25<br /> = <br /> = 1,1914<br />Untuk x = 1yi+1 = yi + f(xi, yi).h<br /> = 1,1914 + f(0,75;1,1914).0,25<br /> = <br /> = 1,3961<br />07/01/2011<br />6<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
7. 7. Metode Euler (Ex.)<br />07/01/2011<br />7<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
8. 8. Metode Euler (Ex.)<br />Nilaieksak<br />07/01/2011<br />8<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
9. 9. Metode Euler (Ex.)<br /><ul><li>Padasaat x = 0; y = 1
10. 10. Persamaan</li></ul>07/01/2011<br />9<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
11. 11. Metode Euler (Ex.)<br /><ul><li>Untuk x = 0,25 </li></ul>07/01/2011<br />10<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
12. 12. Metode Euler (Ex.)<br />Untuk x = 0,5 <br />07/01/2011<br />11<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
13. 13. Metode Euler (Ex.)<br />Untuk x = 0,75 <br />07/01/2011<br />12<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
14. 14. Metode Euler (Ex.)<br />Untuk x = 1 <br />07/01/2011<br />13<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
15. 15. Metode Euler (Ex.)<br />07/01/2011<br />14<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
16. 16. TERIMA KASIH<br />07/01/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />15<br />